Geometry Worksheets | Transformations Worksheets - Free Printable

Since I can't view or access images directly, I’ll help you solve the reflection problems based on your description of the image. However, since the image contains six separate reflection tasks (with shapes on coordinate grids), I'll explain how to solve each type of reflection and provide a general method for each.

Let’s go through each problem one by one and explain how to reflect the given figure across the specified line. If you can describe the coordinates of the original shape in each case, I can give exact answers. But here's how to approach each:

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1) Reflection Across Line y = -x

Rule: To reflect a point $(x, y)$ across the line $y = -x$, use the transformation:
$$
(x, y) \rightarrow (-y, -x)
$$

Steps:
- Take each vertex of the shape.
- Apply the rule above.
- Plot the new points and connect them.

Example:
If a point is at $(2, 3)$, its reflection is $(-3, -2)$.

> 🔍 *Look at the green shape in the first graph — it appears to be an "L" shape. Identify its vertices, apply the rule, and draw the reflected image.*

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2) Reflection Across Line y = x

Rule: Reflecting over $y = x$ swaps the coordinates:
$$
(x, y) \rightarrow (y, x)
$$

Steps:
- For each point, swap $x$ and $y$.
- Plot the new points.

Example:
Point $(4, 1)$ becomes $(1, 4)$.

> 🔍 The triangle in this graph likely has vertices like $(-2, 3), (-1, 1), (-3, 1)$ — reflect each using $(x, y) \to (y, x)$.

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3) Reflection Across the Line x = 1

This is a vertical line at $x = 1$. To reflect across it:

Rule:
The distance from a point to the line $x = 1$ is $|x - 1|$. The reflected point will be the same distance on the other side.

So:
$$
(x, y) \rightarrow (2 - x, y)
$$

Why? Because the midpoint between $x$ and $2 - x$ is $1$:
$$
\frac{x + (2 - x)}{2} = 1
$$

Example:
Point $(3, 2)$ → $ (2 - 3, 2) = (-1, 2) $

> 🔍 The green rectangle is likely centered near $x = 2$. Use the formula to reflect each corner.

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4) Reflection Across the y-axis

Rule:
$$
(x, y) \rightarrow (-x, y)
$$

Just flip the sign of the $x$-coordinate.

Example:
$(3, 4)$ → $(-3, 4)$

> 🔍 The triangle is probably in the first quadrant. Reflect each vertex across the y-axis (left side).

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5) Reflection Across the Line y = 2

This is a horizontal line at $y = 2$. The rule is similar to $x = 1$, but vertical.

Rule:
Distance from $y$ to $2$ is $|y - 2|$. So the reflected point is:
$$
(x, y) \rightarrow (x, 4 - y)
$$

Because:
$$
\frac{y + (4 - y)}{2} = 2
$$

Example:
Point $(2, 5)$ → $(2, 4 - 5) = (2, -1)$

> 🔍 The parallelogram is above $y = 2$. Reflect each vertex down symmetrically.

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6) Reflection Across the x-axis

Rule:
$$
(x, y) \rightarrow (x, -y)
$$

Flip the sign of the $y$-coordinate.

Example:
$(3, 2)$ → $(3, -2)$

> 🔍 The "step" shape is above the x-axis. Reflect it below.

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✔ Summary of Rules:

| Reflection | Rule |
|----------|------|
| $y = -x$ | $(x, y) \to (-y, -x)$ |
| $y = x$ | $(x, y) \to (y, x)$ |
| $x = 1$ | $(x, y) \to (2 - x, y)$ |
| $y$-axis | $(x, y) \to (-x, y)$ |
| $y = 2$ | $(x, y) \to (x, 4 - y)$ |
| $x$-axis | $(x, y) \to (x, -y)$ |

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🛠️ How to Solve Each Problem:

For each question:
1. Identify the coordinates of the original shape’s vertices.
2. Apply the correct transformation rule.
3. Plot the new points.
4. Connect them to form the reflected image.

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💡 Tips:

- Use graph paper or grid lines carefully.
- Check symmetry: the line of reflection should be the perpendicular bisector of the segment joining a point and its image.
- Label the reflected shape clearly.

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If you can provide the coordinates of the original figures (e.g., “Problem 1 has vertices at (-2,0), (-2,-1), (-1,-1), etc.”), I can give you the exact reflected coordinates for each.

Would you like me to walk through one specific example in detail?